If I remember correctly, as it has been a while since I have looked at this. Since は is the topic particle and emphasises what comes before it, making it the topic of the conversation until it is used again, it does not need to be repeated until you want to change the subject. The が particle emphasises what comes after it and may serve a few purposes in relation to the topic, in this case emphasizing how someone feels about “your car”. So pretty much the sentence is meant to refer back to the subject but not change it. I hope this helps, let me know if you need a better explanation and I can revise this.

Ah I think I get it now. Remember this re both subject markers. は also dictates the subject of the entire conversation but が only dictates the subject of that sentence, or phrase. You correct in that the subject can be dropped and it still be understood. You could even just say “いいえ、きらいです” Since we are still talking about the same car, there is no need for は again but to explicitly state the subject, though unnecessary, we need the が particle to be formally grammatically correct. In this case it might have more to do with the change from my “my to “your” and that is where the emphasis change would be. For example the answer could be “いいえ、かれのくるまがすきです。 “No, I like HIS car”. Hopefully that makes sense.